Question

There were 15 million people in 1980 (when t = 0) and 80 million people in 1990. How do you find an exponential model for the population (in millions of people) at any time t, in years after 1980?

Answer 1

Hint: To find the exponential model for the population we are going to use the exponential form which is as follows: $y=u{{v}^{t}}$. In this equation, u and v are constants and we are going to find the value of u and v by applying two conditions in the first condition, we are going to put t = 0 and y = 15 million and in the second condition we are going to put t = 10 and y = 80 million. Using these two equations we will get the value of u and v and hence will get the exponential model of the population.

Complete step by step solution:
In the above problem, we have given 15 million people in 1980 (and we are considering this is as t = 0) and 80 million people in 1990 (and this is the population at t = 10).
Now, to find the exponential model of this population, we are going to use the following exponential form which is as follows:
$y=u{{v}^{t}}$ ………. (1)
Substituting y as 15 and t as 0 in the above equation we get,
$\begin{align}
  & \Rightarrow 15=u{{v}^{0}} \\
 & \Rightarrow 15=u...........(2) \\
\end{align}$
Now, we are going to put y as 80 and t as 10 in eq. (1) we get,
$\Rightarrow 80=u{{v}^{10}}$
Also, from eq. (2), we are going to put the value of u as 15 in the above equation and we get,
$\Rightarrow 80=\left( 15 \right){{v}^{10}}$
Dividing 15 on both the sides we get,
$\Rightarrow \dfrac{80}{15}={{v}^{10}}$
Now, taking $\dfrac{1}{10}$ as power on both the sides of the above equation we get,
$\begin{align}
  & \Rightarrow {{\left( \dfrac{80}{15} \right)}^{\dfrac{1}{10}}}={{v}^{\dfrac{10}{10}}} \\
 & \Rightarrow {{\left( \dfrac{80}{15} \right)}^{\dfrac{1}{10}}}=v........(3) \\
\end{align}$
Now, using values of u and v from eq. (2 and 3) and substitute these values in eq. (1) we get,
\[y=\left( 15 \right){{\left( \dfrac{80}{15} \right)}^{\dfrac{t}{10}}}\]
Hence, we have got the exponential model of the given population as follows:
\[y=\left( 15 \right){{\left( \dfrac{80}{15} \right)}^{\dfrac{t}{10}}}\]

Note: Generally, we think that exponential form is something to the power of e but exponential is some number to the power of some number which we have shown in the above solution.
Generally, we think that exponential model is of the form:
$y=a{{e}^{t}}$
But it is not necessary that the exponential model always contain “e”, it can be any number.